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# Tensor Calculus for Physics

A Concise Guide

Dwight E. Neuenschwander

Publication Date

**Using a clear, step-by-step approach, this book explains one of the more difficult—yet crucial—topics in physics.**

Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. A horizontal electric field producing vertical polarization in dielectrics; an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis; an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts—these are some situations where physicists employ tensors. But the true beauty of tensors lies…

**Using a clear, step-by-step approach, this book explains one of the more difficult—yet crucial—topics in physics.**

Understanding tensors is essential for any physics student dealing with phenomena where causes and effects have different directions. A horizontal electric field producing vertical polarization in dielectrics; an unbalanced car wheel wobbling in the vertical plane while spinning about a horizontal axis; an electrostatic field on Earth observed to be a magnetic field by orbiting astronauts—these are some situations where physicists employ tensors. But the true beauty of tensors lies in this fact: When coordinates are transformed from one system to another, tensors change according to the same rules as the coordinates. Tensors, therefore, allow for the convenience of coordinates while also transcending them. This makes tensors the gold standard for expressing physical relationships in physics and geometry.

Undergraduate physics majors are typically introduced to tensors in special-case applications. For example, in a classical mechanics course, they meet the "inertia tensor," and in electricity and magnetism, they encounter the "polarization tensor." However, this piecemeal approach can set students up for misconceptions when they have to learn about tensors in more advanced physics and mathematics studies (e.g., while enrolled in a graduate-level general relativity course or when studying non-Euclidean geometries in a higher mathematics class).

Dwight E. Neuenschwander's *Tensor Calculus for Physics* is a bottom-up approach that emphasizes motivations before providing definitions. Using a clear, step-by-step approach, the book strives to embed the logic of tensors in contexts that demonstrate why that logic is worth pursuing. It is an ideal companion for courses such as mathematical methods of physics, classical mechanics, electricity and magnetism, and relativity.

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Reviews

This book is well written and has sufficient rigor to allow students to use it for independent study.

An introductory* Tensor Calculus for Physics *book is a most welcome addition... Professor Neuenschwander's book fills the gap in robust fashion.

Dr. Neuenschwander has written the book I wish I could have read when I was learning mathematical physics. It is colloquial yet authoritative, with many treasures lurking within the chapters. The book brings a breath-taking cohesion to the varied ideas about vectors and tensors that are typically spread across the undergraduate’s school terms haphazardly. It is the book I’ll want to have nearby next time I teach special or general relativity, or upper-level electromagnetism or rotational dynamics, as much for the historical tidbits and definitive examples, as for its asides about vocabulary and conventions and for its bibliography.

About

Publication Date

Status

Available

Trim Size

6

x 9

Pages

248

ISBN

9781421415659

Illustration Description

8 line drawings

Table of Contents

Preface

Acknowledgments**Chapter 1. Tensors Need Context**1.1. Why Aren't Tensors Defined by What They Are?

1.2. Euclidean Vectors, without Coordinates

1.3. Derivatives of Euclidean Vectors with Respect to a

Preface

Acknowledgments**Chapter 1. Tensors Need Context**1.1. Why Aren't Tensors Defined by What They Are?

1.2. Euclidean Vectors, without Coordinates

1.3. Derivatives of Euclidean Vectors with Respect to a Scalar

1.4. The Euclidean Gradient

1.5. Euclidean Vectors, with Coordinates

1.6. Euclidean Vector Operations with and without Coordinates

1.7. Transformation Coefficients as Partial Derivatives

1.8. What Is a Theory of Relativity?

1.9. Vectors Represented as Matrices

1.10. Discussion Questions and Exercises

2.2. The Inertia Tensor

2.3. The Electric Quadrupole Tensor

2.4. The Electromagnetic Stress Tensor

2.5. Transformations of Two-Index Tensors

2.6. Finding Eigenvectors and Eigenvalues

2.7. Two-Index Tensor Components as Products of Vector Components

2.8. More Than Two Indices

2.9. Integration Measures and Tensor Densities

2.10. Discussion Questions and Exercises

3.2. Relative Motion

3.3. Upper and Lower Indices

3.4. Converting between Vectors and Duals

3.5. Contravariant, Covariant, and "Ordinary" Vectors

3.6. Tensor Algebra

3.7. Tensor Densities Revisited

3.8. Discussion Questions and Exercises

4.2. The Affine Connection

4.3. The Newtonian Limit

4.4. Transformation of the Affine Connection

4.5. The Covariant Derivative

4.6. Relation of the Affine Connection to the Metric Tensor

4.7. Divergence, Curl, and Laplacian with Covariant Derivatives

4.8. Disccussion Questions and Exercises

5.2. The Riemann Tensor

5.3. Measuring Curvature

5.4. Linearity in the Second Derivative

5.5. Discussion Questions and Exercises

6.2. General Covariance and Gravitation

6.3. Discussion Questions and Exercises

7.2. Metrics on Manifolds and Their Tangent Spaces

7.3. Dual Basis Vectors

7.4. Derivatives of Basis Vectors and the Affine Connection

7.5. Discussion Questions and Exercises

8.2. 1-Forms and Their Extensions

8.3. Exterior Products and Differential Forms

8.4. The Exterior Derivative

8.5. An Application to Physics: Maxwell's Equations

8.6. Integrals of Differential Forms

8.7. Discussion Questions and Exercises

Appendix A: Common Coordinate Systems

Appendix B: Theorem of Alternatives

Appendix C: Abstract Vector Spaces

Bibliography

Index

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